by Ronald G. Douglas
I. Introduction
For me, the year 1978 was a good year! In the winter I delivered the Hermann Weyl Lectures [e3] at the Institute for Advanced Study in Princeton. In the summer I gave an invited talk at the International Congress of Mathematicians held in Helsinki. And in December, I met Paul Baum. I had to navigate the snowy roads between Stony Brook and Princeton, and I lost my luggage on the trip to Helsinki, but the year ended on a good note!
On the plane trip returning from Helsinki, my then-colleague Jeff Cheeger told me I should look up Paul Baum in connection with my work on \( K \)-homology. Earlier that summer, for the same reason, Michael Atiyah had suggested to Paul that he look me up. Both Paul and I had come up with concrete realizations of \( K \)-homology, and Michael Atiyah believed relating the realizations could be very important.
My work had started with a problem in abstract operator theory seeming to have little contact with \( K \)-theory. It was the outcome of this joint effort with Larry Brown and Peter Fillmore, now called BDF theory, about which I spoke in Princeton and Helsinki. An extant write-up can be found in [e2], [e3].
In the 60’s a topic of considerable interest among operator theorists concerned compact perturbations and properties of operators which held modulo them. In particular, a result of Herman Weyl and John von Neumann showed that self-adjoint operators with the same essential spectrum were unitarily equivalent modulo the compact operators. Here, essential spectrum is defined as limit points of the spectrum or eigenvalues of infinite multiplicity. People were interested in the analog of this result for normal operators. And, in particular, Paul Halmos had asked if one could characterize such operators. David Berg had shown that normal operators with the same essential spectrum were unitarily equivalent modulo compacts. But whereas an operator that is self-adjoint modulo the compacts must be a self-adjoint plus a compact, the analogous statement is no longer true for normal operators. The unilateral shift provides a counterexample. In particular, Paul Halmos raised two specific questions. First, when is an operator that is normal modulo the compacts the sum of a normal and a compact? Second, is the collection of sums of a normal operator and a compact operator closed in the norm topology? These are some of the problems that Larry Brown, Peter Fillmore, and I were investigating.
In tackling these questions, following earlier work of Lewis Coburn, we eventually focused on the associated short exact sequence of \( C^* \)-algebras \[ 0\to\mathcal{K}(\mathcal{H})\to\mathcal{E}(T)\to C(X_T )\to0. \] Here \( \mathcal{K}(\mathcal{H}) \) is the two-sided ideal of compact operators on \( \mathcal{H} \), \( T \) is an operator on the Hilbert space \( \mathcal{H} \) satisfying \[ [T,T^* ]=TT^* -T^* T\in\mathcal{K}(\mathcal{H}), \] \( \mathcal{E}(T) \) is the \( C^* \)-algebra generated by \( T \) and \( I \), and \( X_T \)is the essential spectrum of \( T \) or the spectrum of \( \pi(T) \) in the Calkin algebra \( \mathcal{L}(\mathcal{H})/\mathcal{K}(\mathcal{H}) \). We realized that the collection of such extensions formed a commutative semigroup with identity in a natural fashion and that the question of when \( T=N+K \) is equivalent to the question of when the corresponding extension is trivial or splits. But of course, this only replaced one problem with another problem; that is, when is such an extension trivial?
One obstruction to triviality is the existence of a \( \lambda \) not in the essential spectrum of \( T,\,\sigma_e (T) \), for which the Fredholm index of \( T-\lambda \) is not zero. (Recall that an operator is said to be Fredholm if it has closed range and both its kernel and cokernel are finite-dimensional. Moreover, its index is the difference of the dimensions of the kernel and the cokernel.) My results with Larry Brown and Peter Fillmore ultimately showed that the converse held; that is, the extension is trivial if and only if there is no such \( \lambda \). But proving that took some doing.
In considering the analogous question of triviality for spaces \( X\subseteq \mathbb{C}^m \) or for \( m \)-tuples of essentially normal operators \( (T_1,\dots,T_m) \) that commute modulo the compacts, we realized that one needed to consider tensoring the corresponding extension by \( M_m(\mathbb{C}) \). Eventually, this led us to the fact that the semigroup was a group that had a natural pairing with \( K \)-theory. Ultimately, this showed that such extensions yielded a concrete realization for odd \( K \)-homology and that, for \( X \) a subset of the plane, the Fredholm index mentioned above was enough to decide whether or not an extension was trivial.
Following Alexander Grothendieck’s introduction of algebraic \( K \)-theory, Michael Atiyah and Fritz Hirzebruch defined topological \( K \)-theory, and Michael Atiyah later showed that an abstraction of elliptic operators yielded a concrete analytic realization of even \( K \)-homology. However, he was unable to describe the equivalence relation. The BDF work provided a realization of the odd \( K \)-homology group and a workable set of equivalence relations. Using the relation between the odd and even \( K \)-homology groups, one could complete Michael Atiyah’s realization of the even \( K \)-homology. About the same time, Gennadi Kasparov was attempting to follow up on Michael Atiyah’s work to obtain an equivalence relation and use it to study the Novikov conjecture, which was Atiyah’s original motivation.
Paul Baum had started out trying to provide a concrete geometric realization of \( K \)-homology for a space \( X \). After several attempts, he had come up with a picture that Michael Atiyah liked. It involved the notion of a \( \mathrm{spin}^c \)-manifold, a notion known to be related to \( K \)-theory. Let \( X \) be a finite complex. For each group \( K_i (X),\,i=0,1 \), one considers every compact \( \mathrm{spin}^c \) manifold \( M \) with dimension of the same parity as \( i \), every complex vector bundle \( E \) over \( M \), and every continuous map \( f \) from \( M \) to \( X \). The triples \( (M,E,f) \) are the cycles for \( K_i (X) \). Paul Baum had an equivalence relation built from three steps. The critical step was what Paul called “vector bundle modification”. It captures Bott periodicity and the fact that \( K_i (X) \) is periodic in \( i \) with period 2. One might consider Paul’s cycles to be analogous to those in singular homology theory, with \( (M,f) \) playing the role of singular simplex and \( E \) the role of multiplicity.
Michael Atiyah had shown the existence of \( K \)-homology using Spanier–Whitehead duality and had provided a concrete realization for it. But to be useful one needed to be able to recognize naturally occurring cycles for this theory and know when they are equivalent. In BDF we had analytic cycles, while in Paul Baum’s description we had geometric ones. Michael Atiyah’s belief that an explicit isomorphism between these two pictures could be very important led to our meeting.
II. The isomorphism of realizations
At the end of the 70’s, an airline flew from Providence, Rhode Island, to Islip, New York, and back, stopping at New Haven and Bridgeport in Connecticut, before returning to Providence. Paul and I flew those routes regularly, once every month or two for the next year and a half. Initially most of our time was devoted to explaining to each other our realizations of \( K \)-homology. Since each of us had limited expertise in the other’s field, this exercise required considerable effort, and the consumption of much wine and good food, especially at a now-closed restaurant called Napoleon’s in Port Jefferson, New York.
In Michael Atiyah’s realization of even \( K \)-homology, \( K_0 (X) \), for \( X \) a finite complex, he defined cycles in terms of an abstract notion of an elliptic operator: \( (T,\sigma_1 ,\sigma_2 ) \), where \( \sigma_1 \) and \( \sigma_2 \) are \( * \)-representations of \( C(X) \) on Hilbert spaces \( \mathcal{H}_1 \) and \( \mathcal{H}_2 \) and \( T \) is a Fredholm operator from \( H_1 \) to \( H_2 \) which intertwines \( \sigma_1 \) and \( \sigma_2 \) up to the compacts. The covariance required of a homology theory arises from the following construction. Let \( f : X \rightarrow Y \) be a continuous map and let \( (T,\sigma_1 ,\sigma_2 ) \) be an abstract elliptic operator on \( X \). Define \( f^* : C(Y) \rightarrow C(X) \) by \( f^* (\xi ) = \xi \circ f \). Then \( (T,\sigma_1 \circ f^*,\sigma_2 \circ f^* ) \) is an abstract elliptic operator on \( Y \). If \( X \) is a compact manifold, if \( T \) is given by an elliptic pseudodifferential operator of order 0 between sections of vector bundles \( E_1 \) and \( E_2 \) over \( X \), and if continuous functions on \( X \) act on sections of these vector bundles by pointwise multiplication, then one obtains an “Atiyah cycle” from this data, and Michael Atiyah showed there were enough of those cycles to generate \( K_0 (X) \). However, he was unable to describe explicitly the necessary equivalence relation.
Paul and I realized that the same setup with \( \mathcal{H}_1 = \mathcal{H}_2 \), \( \sigma_1 =\sigma_2 \), and \( T \) a self-adjoint Fredholm operator enables one to obtain an odd cycle following BDF. In particular, if \( P \) is the orthogonal projection onto the positive spectral space for \( T \), then compressing a multiplier \( \phi \) in \( C(X) \) by \( P \) to obtain a “Toeplitz-like operator”, \( P\sigma(\phi)P \), one obtains a unital \( * \)-homomorphism from \( C(X) \) to the Calkin algebra of the positive spectral subspace, which is another version of a BDF cycle. In particular, this showed that the odd analog of the Atiyah–Singer index theorem could be viewed as the classical index theorem for Toeplitz operators such as those defined on the unit circle by Israel Gohberg and Mark Krein. This observation also enabled us to define the sought-after isomorphism between the analytic and geometric realizations of \( K \)-homology.
Recall that for a finite-dimensional Hilbert space \( G \), one can define the associated Clifford algebra \( \mathrm{Cl}(G) \) on which \( G \) acts by multiplication. This action is not irreducible since \( \mathrm{Cl}(G) \) splits into an orthogonal direct sum of submodules. Let \( M \) be a smooth closed manifold with a Riemannian metric. The fibers of the cotangent bundle, \( T^*M \), to \( M \) can be used to define a bundle of complex Clifford algebras \( \mathrm{Cl}(T^*M) \) over \( M \). The bundle \( T^*M \) acts on \( \mathrm{Cl}(T^*M) \) but not irreducibly. The statement that \( M \) is a \( \mathrm{spin}^c \) manifold means that not only does one have this pointwise action on sections of the cotangent bundle but there is a sub-bundle of the Clifford bundle \( \mathrm{Cl}(T^*M) \), called the spinor bundle, on which \( T^*M \) acts irreducibly. This enables one to define the Dirac operator, \( D_M \), on smooth sections of this spinor bundle to obtain a first-order elliptic differential operator. When \( M \) is odd-dimensional, the Dirac operator is self-adjoint and the construction described in the preceding paragraph, applied to \( D(1 + D^2)^{-1/2} \), yields an odd BDF cycle. Moreover, if we tensor \( D \) with the identity operator \( I_E \) on a vector bundle \( E \), we can proceed as above to make from \( D\otimes I_E \) a BDF cycle constructed from the pair \( (M,E) \). A choice of connection on \( E \) is required, but the resulting odd cycle defines a class in \( K_1(M) \) that is independent of the choice of connection. Call this class \( [D\otimes I_E] \). For a given triple \( (M, E, f) \) with \( \dim M \) odd, using the covariance defined by composing the \( C(M) \)-representation with \( f^* \), we can map \( [D\otimes I_E] \in K_1(M) \) to a class \[ f_*([D\otimes I_E]) \in K_1 (X) \] that is defined by a BDF cycle associated with an abstract elliptic operator on \( X \). The map \[ (M,E,f)\mapsto f_* ([D \otimes I_E] ) \] yields the desired isomorphism between geometric and analytic odd \( K \)-homology.
In the setting of the preceding paragraph, but with \( M \) even-dimensional, the bounded operator constructed from \( D\otimes I_E \) decomposes to \begin{equation*} \begin{pmatrix} 0 & T^* \\ T & 0 \end{pmatrix} \end{equation*} relative to the decomposition of the spinor bundle into even and odd spinors. Using \( T \), we obtain an Atiyah cycle on \( M \) and we can push its class into \( K_0 (X) \), as before, to obtain the map \[ (M,E,f)\mapsto (T,\sigma_0 \circ f^*, \sigma_1 \circ f^*), \] where the \( \sigma_i \)’s are the representations of \( C(M) \) on the sections of \( M \)’s even and odd spinor bundles. This map defines the desired isomorphism in the even case.
By the time Paul and I attended the AMS Summer Institute in Operator Algebras in Kingston, Ontario, in the summer of 1980, we had understood the isomorphism and how to establish it. It is closely related to the Atiyah–Singer index theorem. These results were described in [1].
The Kingston meeting was Paul’s debut as an “operator algebraist”. After that he was no longer able to be anonymous in that community. He drove from Providence, took the ferry to Long Island, and picked me up at Stony Brook on the way to Kingston. He was one of the “geometer-topologists” who got interested in what is now known as noncommutative geometry. Alain Connes was at the Kingston meeting also.
III. Relative \( K \)-homology
I had planned to visit Alain Connes in Paris for the fall semester. As it turned out, I could make it for only a week and Paul joined me. During that time, Paul and Alain started work on what has become known as the Baum–Connes conjecture. Paul and I, on the other hand, started working on understanding an analytic realization for the even relative \( K \)-homology. Some motivation for this was provided by the proof of the Atiyah–Singer index theorem given in the Palais notes [e1]. We got together frequently over the next two or three years, working to understand this realization. We also worked to understand the relation of our work to that of Gennadi Kasparov, who was attempting to complete Michael Atiyah’s original effort to resolve the Novikov conjecture. In his work, a two variable “\( KK \)-theory” appeared and he had incorporated the \( C^* \)-extension framework.
In defining \( K \)-homology theory there are essentially four groups: the even group \( K_0 (X) \), the odd group \( K_1 (X) \), the even relative group \( K_0 (X,A) \), and the relative odd group \( K_1 (X,A) \), where \( X \) is a compact metrizable space and \( A \) is a closed subset of \( X \). While it is possible to define the relative groups abstractly in terms of the absolute groups, we believed that identifying a concrete realization of relative cycles would lead to applications and new connections. Our first announcement of what we thought to be true was given at the U.S.-Japan operator algebras meeting held in Kyoto in the summer of 1983. For Paul and for me this was a first visit to Japan. On the way to Kyoto I stopped at Sendai at the invitation of Masamichi Takesaki before continuing to Kyoto. In Kyoto we stayed at a Holiday Inn and we recounted our adventures, both mathematical and otherwise, during breakfast each morning. Paul was particularly intrigued by the Japanese restaurants and the “geisha-like waitresses”. There was also a cafe on the roof at which “post-World War II Hawaiian songs” were the order of the day. The conference enabled us to discuss our relative \( K \)-homology program with other participants at the workshop.
There are two parts to the realization of even relative \( K \)-homology: one for the cycles in \( K_0 (X,A) \) and a second for the boundary map \( \partial:K_0 (X,A)\to K_1 (A) \). To be useful the latter map needs to be very explicit. The even relative cycle is a variation of the absolute Atiyah cycle \( (T,\sigma_1 ,\sigma_2 ) \) in which \( T \) is now assumed only to be semi-Fredholm, which means the kernel of \( T \) can be infinite-dimensional. Additionally, one assumes that for \( \phi \in C(X) \) the compression of a multiplier \( \sigma_1 (\phi) \) to the kernel of \( T \) is compact if \( \phi \) vanishes on \( A \). This allows one to define the \( K_1 \) cycle that is the image of the boundary map as follows. Take \( \xi \) in \( C(A) \), extend it arbitrarily to \( X \) to obtain \( \phi \) in \( C(X) \), and compress multiplication by \( \sigma_1 (\phi) \) to the kernel of \( T \). The resulting operator is well-defined up to compacts and, with \( Q \) denoting projection onto the kernel of \( T \), the map \( \xi \mapsto Q \sigma_1 (\phi)Q \) defines a unital \( * \)-homomorphism from \( C(A) \) to the Calkin algebra of the kernel of \( T \); i.e., it defines a BDF cycle representing a class in \( K_1 (A) \). This cycle represents the image of \( (T,\sigma_1 ,\sigma_2 ) \)’s class under the desired boundary map \( \partial:K_0 (X,A)\to K_1 (A) \).
Showing that this definition yields the relative even group along with the boundary map was tackled by Paul and myself during the special year 1984–85 at MSRI in Berkeley. Part of the proof rested on matching up these groups with corresponding \( KK \)-groups of Gennadi Kasparov. The program at MSRI was organized by Alain Connes, Masamichi Takesaki, and myself. Because Paul had recently moved from Brown University to Penn State, he was unable to join the group for the whole year. However, he was a frequent visitor for several weeks at a time.
IV. Differential operators and the relative even group
Although our work provides a rather explicit description of relative even cycles and the boundary map, the description is not terribly useful in the form given above. The problem is to decide how one recognizes a relative cycle and the image of the boundary map when the cycle arises from a differential operator on a smooth manifold. To accomplish this, we were joined by Michael Taylor, who at that time was my colleague at Stony Brook and who is an expert on PDEs. We focused on differential operators, and a generalization of them, called pseudodifferential operators, defined from smooth sections of a vector bundle \( E_0 \) to smooth sections of a vector bundle \( E_1 \) on a smooth manifold \( M \) with boundary \( \partial M \).
Since the differential operator defines an unbounded operator on the space of smooth sections, one has to proceed carefully. In particular, one must consider extensions of this operator that are closed. We showed that some of them define even relative cycles. Because such operators have more than one possible closed extension, the issue of uniqueness arises, but under broad hypotheses it turns out that the class represented by the relative cycle is independent of the choice of closed extension. One can also identify the odd boundary cycle explicitly, now as an elliptic pseudodifferential operator \( D_\partial \) on \( \partial M \). The correspondence \[[D]\to[D_\partial ]\] defines the boundary map from \( K_0 (M,\partial M) \) to \( K_1 (\partial M) \).
The fact that different closed extensions of \( D \) define the same element of \( K_0 (M,\partial M) \) implies that their corresponding images, under the boundary map, in \( K_1 (\partial M) \) are equal, which has far-reaching consequences. For example, if \( D \) is the Dirac operator \( D_M \) on \( M \) (where we assume \( M \) has a \( \mathrm{spin}^c \) structure) then the image of \( [D_M ] \) is the class defined by the Dirac operator on \( \partial M \) defined by the \( \mathrm{spin}^c \) structure on \( \partial M \) induced by the \( \mathrm{spin}^c \) structure on \( M \). If one takes the maximal closed extension of \( D_M \), this image under the boundary map is the odd cycle defined by the Calderón projection. This identification yields the Toeplitz index theorem of Boutet de Monvel and, in fact, allows one to generalize it. Many more results involving index theory follow from this setup.
This realization is particularly interesting in the case \( X \) is a strictly pseudoconvex complex manifold and \( D \) is the \( \overline{\partial} \)-operator. In this case, one can identify the kernel \( \overline{\partial} \) as the finite direct sum of subspaces of mixed forms with the first summand being the Bergman space on \( X \) and with other summands, which involve higher-degree forms, being finite-dimensional. Thus, the Bergman space on \( X \) is identified up to a finite-dimensional subspace with the kernel of the relative even \( K_0 (X, \partial X) \) cycle. If one uses the above recipe to calculate the boundary map, one finds that \( [\overline{\partial}_{\partial X}]=[D_{\partial X}] \). This enables one to conclude that the cycles defined by the Bergman space on \( X \) and the Dirac operator on \( \partial X \) represent the same class in \( K_1 (\partial X) \). Since the Atiyah–Singer index theorem or the isomorphism that we obtained earlier describes the odd cycle defined by the Dirac operator, one is able to calculate the odd cycle defined by the Bergman space. This is essentially the result of Boutet de Monvel at this level of generality, with a proof slightly different from the one mentioned above.
This work, which was published in [2], was essentially completed for the 1988 AMS Summer Institute held in New Hampshire and organized by William Arveson and myself. Many of the themes of the subject now called noncommutative geometry were discussed there as well as at the meeting in Bowdoin, Maine, that followed. The location of the meetings was chosen to take advantage of the summer weather, but mother nature thwarted us. It was hot and fans were in short supply. However, one consequence of these meetings being held in New England was the opportunity to meet Paul’s father, an artist. I had already met his mother, also an artist, on several occasions because she lived in New York City. This opportunity provided some insight into Paul’s personality.
V. The odd relative group and concluding comments
It is also of interest to consider the odd relative group. Here the
relative cycles would be pairs \( (T,\sigma) \), where \( T \) is a symmetric
semi-Fredholm operator on a Hilbert space \( \mathcal{H} \) and \( \sigma \) is a
\( * \)‑representation of \( C(X) \) on \( \mathcal{H} \).
Again, one assumes that for a multiplier \( \sigma (\phi ) \) arising from a
\( \phi \in C(X) \) that vanishes on \( A \) the compression of \( \sigma (\phi ) \)
to the kernel of \( T \) is compact. One needs also to define the boundary map
effectively and the question is somewhat delicate, involving the classical
theory of extensions of
symmetric operators. Subsequent work by Michael Taylor finessed these
difficulties by taking a product \( X\times S^1 \) but at a cost. In particular,
the boundary map is less explicit, as is matching up the boundary map in
the case of differential operators. Although
we discussed how to complete this development directly, we never carried it
out. Again, there are possible applications that I related to Sturm–Liouville
theory in higher dimensions.
It had been ten years at this point since I met Paul Baum. Although our research interests diverged subsequently, we remain good friends and often reminisce about our roles in the development of \( K \)-homology.
